Ariat makes both leather and rubber soled boots. I don’t know anyone wearing rubber soles whether it be Ariat or custom made.
If you want to wear rubber soles, I suggested riding in a wider treaded stirrup. In cold weather(-0F and colder) I wear snow packs but I switch out to oversized stirrups made for overboots and ride with my toes mostly because I’ve been hung up in them.
No it’s not weird to say a lack of slick soles can cause a hang up. Yall that ride English and use sticky treads in your stirrups. If it holds your foot in that much it makes it that much harder to get it out especially in a wreck. If it creates traction it’s not going to help you get loose. I’ve gotten rid of boots(Anderson Beans specifically) that would hang on the waist of the sole making a quick dismount.
Going back to matching boot/heel to stirrup- A bronc rider uses a riding heel or a tall under slung heel with an oxbow or bronc stirrup. It keeps them from blowing a stirrup but yet keeps them from running their whole foot through it.
I ride cutting and cow horses, most of our saddles have a 2" flat bottom stirrup and generally the foot ends up very close to the home position working a cow. A slick bottomed boot with at the very least a roper heel to a riding or underslung heel is usually used.
Cowboying, I ride a 5-6" tread. They tend to kick your feet out of the stirrup much like a lot of the tapaderos sold that are unbalanced. You can get away with tennis shoes if you wanted to for that reason. Comfortable to ride all day in and easy to stand in to rope.
Loose enough boots is another way to protect yourself. If you do get hung up despite doing your best, I was taught to rollover on my belly and twist out of my boots. I can verify it has saved me and holding a horse down on me after a fall to set myself up to twist out as I let him up.
Again I don’t care if you want to use safety stirrups or even rubber soled boots. I gave you additional, inexpensive ways to help yourself from a western perspective. You don’t have to buy expensive safety stirrups You can use inexpensive shelf boots matched with the right stirrups to limit your risk if you’re a casual western rider.